Two point function, hydrody-namics and spectral function

Kwanghyun JoHanyang Univ.

2011. 2. 28 Kps Meeting

Index

1. AdS/QCD 2. Computing Correlators in AdS/CFT 3. Hydrodynamics

- Perturbation in RN AdS5 back-ground

4. Spectral function- Photoemission rate

4. Remarks

AdS/QCD

AdS/CFT : AdS5 X S5 ~ N=4 SYM in 4 dim.

AdS/QCD : aAdSd ~ low energy Yang-Mills in d-1 dim.

Phase transition in AdS/QCD

T Field theory side Gravity side

T<Tc

Hadron Thermal AdS

Tc Confinement/deconfinement phase transition.

Hawking – Page transtition

T>Tc

Quarks and Gluons AdS B.H.

T Field theory side Gravity side

T<Tc

Vev is not zero Thermal AdS

Tc Chiral symmetry breaking/restoration phase transition

Hawking – Page transtition

T>Tc

Vev is zero AdS B.H.

Assume that confinement and chiral phase transition occurs at same scale, Tc.

SU(2)L X SU(2)R flavor sym. -> SU(2)V

Two point function

ZgaugeGenerating functionalLow energy effective theory

(E < Λc) : Chiral Lagrangian, Walecka

Microscopic theory

(E > Λc) : Yang-Mills

perturbative

Non perturba-tive

𝛿2 𝑍𝑔𝑎𝑢𝑔𝑒

𝛿𝜑𝑖𝛿𝜑 𝑗

=𝐺𝑖 𝑗

Intermediate energy regime

Two point function

ZstringGenerating functionalLow energy effective theory

(L > ls) : Einstein gravity

Microscopic theory

(L < ls) : string theory

perturbative

Non perturba-tive

𝛿2 𝑍 𝑠𝑡𝑟𝑖𝑛𝑔

𝛿𝜑𝑖 𝛿𝜑 𝑗

=𝐺𝑖 𝑗

Computable regime

Two point function

𝛿2 𝑍𝑔𝑎𝑢𝑔𝑒

𝛿𝜑𝑖𝛿𝜑 𝑗

=𝐺𝑔 , 𝑖 𝑗

𝛿2 𝑍 𝑠𝑡𝑟𝑖𝑛𝑔

𝛿𝜑𝑖 𝛿𝜑 𝑗

=𝐺𝑠 ,𝑖 𝑗

ZgaugeGenerating functional

ZstringGenerating functional

AdS/CFT

Non perturba-tive

Strongly coupled gauge theory

weakly coupled gravity theory

How to compute Green func-tion

Find : , / Perturb : +δ ,

Find and solve the equations of motion for δ,

Differentiate Seff with respect to the source field

plug it into the original action :

Seff

Correlators

Holography : Bulk(d+1 dim) projected onto the Boundary(d dim)

th/0205051

AdS/CFT says, on-shell partition function of Grav-

ity(Bulk) = CFT partition function(Boundary)

solving Bulk e.o.m. and integrating extra dim of bulk action : Boundary action (On-shell action).

Boundary conditions

2nd order linear differential e.o.m.,- two integration constants(boundary conditions)

BH(u=1

)

Bound-ary

(u=0)

(1- u )𝑖𝑤 /4𝜋 𝑇

(1- u )− 𝑖𝑤 /4 𝜋𝑇

Black hole eat every-thing!

Bottom up approach

Five dimensional action : Einstein + Maxwell + cosmological constant

Hadron physics

meson, baryon

π, ρ, etc…

μB, T

hQCD

meson, baryonδ, , , etc…

Non trivial back-ground

Fluctua-tion

But T,There is no hadrons in hQCDbut quark-gluon plasma.

Recipe for Green func-tion

EOM for perturbed fields : δ,

On-shell actionand Green function

Hydrodynamics

Hydrodynamics describes the system at large distance and time scale : (small w, k limit).

Basic equation : conservation of energy momentum

Hydrodynamics

Transport coefficient : viscosity, thermalization time, vorticity etc.

Hydrodynamics

Transport coefficient : viscosity, thermalization time, vorticity etc.

+ O(w,q)

Taylor expansion of the equation of motion in w, q near 0.The solution of each order in w, q is the , .

Hydrodynamics: fundamental d.o.f. = densities of conserved charges

Need to add constitutive relations!

Example: charge diffusion

[Fick’s law (1855)]

Conservation law

Constitutive relation

Diffusion equation

Dispersion relation

Expansion parameters:

Hydrodynamics

Viscosity/entropy ratio in QCD: current status

QCD: RHIC elliptic flow analysis suggests

Theories with gravity duals in the regimewhere the dual gravity description is valid

(universal limit)

QCD: (Indirect) LQCD simulations

H.Meyer, 0805.4567 [hep-th]

Trapped strongly correlated cold alkali atoms

Liquid Helium-3

T.Schafer, 0808.0734 [nucl-th]

Kovtun, Son & A.S; Buchel; Buchel & Liu, A.S

Hydrodynamics, ex1)

EOM

Solution

Greens function

Black D3 metric and mode ( )

Kubo formu-lar

Shear viscos-ity

Eta/s

Hydrodynamics, ex2)

EOM

Solution

Greens function

Black D3 metric and R charge current δA0 mode

R-charge Diffusion con-stant

Hydrodynamics, ex3)

EOM

Solution

Greens function

Black D3 metric and mode,

Bulk viscos-ity

Momen-tum diffu-sion con-stant

Spectral function

Imaginary part of Green function.

Maximum value 1/at p=m

Correlator (Finite tempera-ture)

N-point function = differentiating N times partition function

with respect to the source

Infalling condi-tion

Near hori-zon

Near boundary

Local series solutions : two indicial solu-tions

(2nd order differential equations)

Spectral function (Im G)

q =0.3

Im GxtxtIm Gxx

q = 0.5

q =0.7

In medium effects of QCD

QCD phase dia-gram

Particle produc-tion In RHIC

In medium effects of QCD

Theory has pa-rameters

Hadron physics hQCD

T (temperature)μ (chemical poten-tial)

Q (black hole charge)

The w, k is scale by tempera-ture

By changing μ, we will see the density effect on hydrodynamic quantities and spectral function

RN AdS5

Metric

Temperature

Chemical pot.

SO(p-1) classification

t x y z r

hMN =

ahtt htx

hai

hμr

hxt hxx

i hia

hyy hij

hji hzz

r hrμ hrr

a=t,x i=y,z

𝑥𝑀=(𝑥 μ ,𝑟 )= (𝑥𝑎 ,𝑥 𝑖 ,𝑟 )

Tensor hij

Vector hai

Scalar hab

AdSp+2 -> p+1 CFT

SO(p) rotational sym.---(wave propagat-ing)-->SO(p-1) rotational sym.hai : bi-vector SO(1,1) X SO(p-1)

Metric and gauge field perturbation: vector mode

Linearized Einstein eq.

Metric ( g = g0 + h ) + gauge field ( A = A0 + a )

With finite density h and a is coupled via background field g0 and A0 .

decoupling, master vari-able

1005.0200

<JxJx> Correlator

Near boundary series solution of master variable

From the definition of master variables, get the transformation matrix R to convert original boundary value and their conjugates

Photoemission rate th/0607237

# of emitted photon / Vol4Leading in e^2 ex-pandsion(e is the EM coupling)Wightman function of EM currents (not

time-ordered)

Other way to get the Wightman function

Photo emission rate

Photoemission rate3 flavor massless QCD (dashed curve) / SYM (Solid Blue)

αSYM = .025, αs = .1

αSYM = .011, αs = .1

Matching Asymptotic fermion mass

Matching Debye screening mass

th/0607237ph/0111107

SYM computation gives quite good hint of hot QGP

Photoemission rate

N=4 Super Yang-Mills computation with small t’ Hooft coupling

Solid Black (SYM : λ=∞)Dashed Blue (λ = 0.5)Dotted Red (λ = 0.2)

th/0607237

Small fre-quencyCoupling in-dep.

𝑘1/2 𝑘2 /3

Known large k asymptote

Photo emission rate

Photo emission rate with unit Maximum at

Maximum rate is decreased first and in-creased after

0μ=10

5

1

Remarks

Holography : two point function of QGP

Small w, k limit : Hydrodynamics - transport coefficients Full w, k regime : Spectral function

- quasi particle peaks, photoemission Finite T and μ, RN AdS is holographic

dual. Thermal photon production is af-

fected by density effects.